A ratio is a statement of a mathematical relationship comparing two quantities, often represented as a fraction. If we consider an angle $\theta$θ in a right-angled triangle, we can construct various ratios to compare the lengths of the sides. Special relationships that exist in right-angled triangles are called trigonometric ratios.

Through investigation we can see that there is a definite relationship between the angles in a right-angled triangle and the ratio of sides. We can use these trigonometric ratios to find unknown angles and sides of a triangle.

There are 3 basic trigonometric ratios that relate sides and angles together. They have the special names of tangent, sine and cosine. The names of the relationships date back to 499AD, with ties to Latin and Sanskrit. Actually the word sine is thought to be derived from a translation gone wrong! Regardless the names for these relationships have stuck.

We often shorten the names tangent, sine and cosine to tan, sin and cos respectively.

Trigonometric ratios

$\sin\theta$sinθ = $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse = $\frac{b}{c}$bc

$\cos\theta$cosθ = $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{a}{c}$ac

$\tan\theta$tanθ = $\frac{Opposite}{Adjacent}$OppositeAdjacent = $\frac{b}{a}$ba

Here is a picture of the above relationships, and for some people the mnemonic of SOHCAHTOA at the bottom is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.

Sin and Cos

For a particular angle in a right-angled triangle, the sin ratio compares the length of the side opposite the angle to the length of the hypotenuse.

The cos ratio compares the length of the side adjacent the angle to the length of the hypotenuse.

Consider the following triangles. $\triangle MOP$△MOP is an enlargement of $\triangle MNQ$△MNQ. This means the angles REMAIN THE SAME between the two triangles.

Consider angle A	In $\triangle MNQ$△`MNQ`	In $\triangle MOP$△`MOP`
Adjacent	4	8
Opposite	3	6
Hypotenuse	5	10
sin A ($\frac{opposite}{hypotenuse}$`oppositehypotenuse`)	$\frac{3}{5}$35	$\frac{6}{10}=\frac{3}{5}$610=35
cos A ($\frac{adjacent}{hypotenuse}$`adjacenthypotenuse`)	$\frac{4}{5}$45	$\frac{8}{10}=\frac{4}{5}$810=45

Overlapping the two triangles:

So even though the side lengths in the two triangles are different:

- the cos ratio of angle A ( $\frac{adjacent}{hypotenuse}$adjacenthypotenuse) remains exactly the same, and

- the sin ratio of angle A ( $\frac{adjacent}{hypotenuse}$adjacenthypotenuse) remains exactly the same.

Later on we will be able to use this property (that every angle has a fixed sin and cos ratio) to find unknown angles.

This applet will let you explore different trigonometric ratios, particularly which sides are needed for a particular ratio. If you would like to see this applet in action, watch this video.

Created with Geogebra

Another relationship

Let's have a look at just one more special relationship.

If we know the sine and cosine ratios for a particular angle,

$\sin\theta=\frac{Opposite}{Hypotenuse}$sinθ=OppositeHypotenuse

$\cos\theta=\frac{Adjacent}{Hypotenuse}$cosθ=AdjacentHypotenuse

Then we can construct a new relationship for $\text{sine }\div\text{cosine }$sine ÷cosine

$\frac{\sin\theta}{\cos\theta}$`sinθcosθ`	$=$=	$\frac{\left(\frac{Opposite}{Hypotenuse}\right)}{\left(\frac{Adjacent}{Hypotenuse}\right)}$(`OppositeHypotenuse`)(`AdjacentHypotenuse`)
	$=$=	$\frac{Opposite}{Hypotenuse}\times\frac{Hypotenuse}{Adjacent}$`OppositeHypotenuse`×`HypotenuseAdjacent`
	$=$=	$\frac{Opposite}{Adjacent}$`OppositeAdjacent`
	$=$=	$\tan\theta$`tanθ`

Algebraically we have just shown that $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ

That is, the tangent ratio of an angle is the same as dividing its sine ratio by its cosine ratio.

Let's have a look at some worked examples.

Question 1

Write down the ratio of the sides represented by $\sin\theta$sinθ.

A right triangle labeled with vertices A, B, and C. The right angle is located at vertex C as indicated by the small square. Side AB is opposite vertex C, and is the hypotenuse. Each of the non-right angles is marked with a small arc near the vertex. The triangle has two angles labeled: angle $\alpha$`α` at vertex A and angle $\theta$`θ` at vertex B. Side BC is adjacent to angle $\theta$`θ`. Side AC is opposite angle $\theta$`θ`.

Question 2

Write down the ratio represented by $\cos\theta$cosθ.

A right-angled triangle labeled with vertices A, B, and C is shown. The angle at vertex A is labeled $\alpha$`α`. The angle at vertex B is labeled $\theta$`θ`. The angle at vertex C is a right angle as indicated by a small square. Opposite vertex C is side AB, which is the hypotenuse. Side BC is adjacent to angle $\theta$`θ`. Side BC is opposite angle $\alpha$`α`. Side AC is adjacent to angle $\alpha$`α`. Side AC is opposite angle $\theta$`θ`.

Question 3

Question 4

Consider the following triangle.

A right-angled triangle with vertices labeled $A$`A`, $B$`B`, and $C$`C`. Each of the non-right angles at vertices $A$`A` and $B$`B` is marked with a small arc near the vertex. Vertex $C$`C` has a right angle, indicated by a small square corner. An angle $\theta$`θ` is indicated at vertex $B$`B`. Side $BC$`BC` is labelled $5$ and is adjacent the angle theta. Side $AC$`AC` is labelled $12$ and is opposite the angle $\theta$`θ`. Side $AB$`AB` is the hypotenuse and labelled $x$ .

Find the value of $x$x.
Hence find the value of $\sin\theta$sinθ. Express your answer as a simplified fraction.
Hence find the value of $\cos\theta$cosθ. Express your answer as a simplified fraction.

Outcomes

10.T.IT.1

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.

Identify sine, cosine and tangent ratios

Trigonometric Ratios